Abstract: A hollow rotor shaft for an electric machine rotor rotating about a longitudinal axis includes: a cylinder shell surrounding a shaft cavity; and end flanges on both shell ends transitioning into respective shaft journals. One end flange has an inlet for leading a cooling medium into the cavity and onto the shell inner surface. Inside the cavity, guides distribute cooling medium entering via the inlet over the inner surface, the shell having at least one cooling-medium outlet opening. The shaft is formed in one piece from a tubular initial body. The guides are raised contours formed from the inner surface and protruding into the cavity. The shaft is flow formed by pressing the raised contour out of the inner surface. The end flanges with respective shaft journals are formed by pressing the two initial body ends. The at least one cooling-medium outlet opening is formed in the raised contour.